Conditional expectation solution-verification

Question

I was wondering if this chain of equalities is valid for i.i.d. random variables $X,Y$ with density function $f$ and let $K$ be any function.

$$E[K({X-Y})] \overset{1}{=} E\big[E[K(X-Y)|Y]\big] \overset{2}{=} E\left[\int_{\mathbb{R}} K(x-Y)f(x)dx\right].$$

Answer 1

Yes, this chain of equalities is valid, provided that $K$ is measurable and that $K(X-Y)$ is integrable. The first inequality would be valid without the assumption of independence. The second one uses the explicit expression of $E\big[E[K(X-Y)|Y]\big]$.